What happens when data is written to a Solid State Disk is worthy of several articles (good summary here), because it is very complicated and depends on the underlying technology. The short story is that SSDs in general cannot write zero bits to memory. Instead, they have to zero out (erase) a whole section of memory, and then they can store data after that by just writing the ones to it. Typically these days they write blocks of 512 bytes but erase a page of 8 blocks which is 4096. This, and the fact that each write/erase cycle causes some physical wear of the memory and the memory does eventually wear out, make SSDs very different than spinning magnetic HDDs.
Setting that aside, SATA drives (and AFAIK SAS drives) do not implement a native command to copy data from one sector to another. (Or at least nothing in the SATA or SAS spec requires them to, so the OS cannot count on such a command being available.) So a file copy across a partition will involve reading the data from one drive sector into host memory and then writing it back out to the drive at a different sector.
This is because as far as the OS is concerned, a drive is a set of numbered logical sectors, and all it can do is read from sectors and write to sectors. The OS cannot tell the drive to remap sectors.
Furthermore, the file system (HFS+, NTFS, ext3, etc.) is a set of data structures that impose order on a set of logical blocks. Those data structures implement "files", "file names", "directories", "permissions", etc. So, yes, when you move a file from one directory to another, it is not copied; only the file system data indicating which directory the file is in gets updated.
The concept of a partition is that it is a set of logical sectors on the drive claimed by a single file system. The corollary to that is that a file system may not access sectors outside of its partition. In large part this is a safety feature, but it also flows from the fact that the file system's data structures are all built around accounting for every sector of the drive under the file system's ownership, and it is non-trivial to add or remove sectors to those structures. This is why you have to run special routines to adjust the size of a partition and also why file systems insist on running on a contiguous set of sectors.
So it is impractical and dangerous to implement a file copy as just transferring sectors from one file system to another. On a spinning magnetic drive, it would also be a performance nightmare, because although the drive will make exceptions for bad sectors, in general it arranges for sectors to be physically located in such a way as to optimize the read and write speed of consecutively numbered sectors.
Additionally, 2 file systems may not store file data the same way on disk, meaning swapping sectors would not work even if it were practical. Even if they are the exact same file system types, say NTFS, one may be using encryption or compression and the other not, or both may encrypting the data, but with different keys. It is not a requirement that the data in the file is exactly what is stored on the disk, all that has to be stored is a reversible transformation of the data, so that the file system can get the data of the file by doing something with the data on disk. So unless both file systems are using exactly the same transformation, simply swapping sectors would not accomplish the goal of transferring the file data.
For all these reasons, it is just too much work for too little gain for the OS writers and file system writers to implement a feature that optimizes moves across partitions for SSDs. So any cross-partition move will be a read and a write.
Inside the SSD, it is a slightly different story. Although the OS did not tell the drive it is copying data from one place to another, writes to SSDs are so expensive (and complicated) that SSD controllers do a lot of work to minimize writes. Some SSDs go so far as to try to detect when a sector being written to storage matches a sector already stored and mark that physical piece of memory as now mapping to 2 different logical sectors rather than copying it, doing at the internal drive level what the OS could not.
But don't count on it.